The technical problem addressed is how to adapt a template dataset so that it matches a target dataset better. The need to adapt a template dataset to better match a target dataset arises routinely in geophysical data processing, analysis, and interpretation. For example, geophysical data contain several types of interfering and undesirable patterns (termed “noise”). The presence of such noise patterns makes it difficult to identify geologic features (termed “signal”) of interest such as those that may contain hydrocarbons. Attenuating the interfering noise patterns in geophysical data improves the detectability and/or interpretability of geologic features. Templates for the noise patterns can often be constructed by using prior knowledge of signal and noise in conjunction with the physics of seismic wave propagation. Since the knowledge about signal and noise that is incorporated during the template construction is never perfect, the predicted templates invariably contain errors. Hence the noise patterns in the target dataset cannot be attenuated by straightforward subtraction of the predicted noise templates. First, the predicted noise template needs to be adapted to overcome the spatially and temporally varying prediction errors and to better match the corresponding noise features in the target dataset. Then the noise in the target dataset can be attenuated by subtracting the adapted noise template.
The amount by which the template can be allowed to change during adaptation (that is, the level of adaptation) is typically problem dependent. Hence the adaptation always needs to be controlled with some constraints. If the adaptation is not adequately constrained, then some features in the adapted noise template would also match the signal components of the target dataset (i.e., overfitting), which is undesirable. For example, in the noise attenuation application described above, looser constraints during adaptation would provide better noise subtraction, but may also cause more damage to the signal. In the subtraction example, the constraints imposed during the adaptation balance the unavoidable tradeoff between noise subtraction and signal preservation. In general, the constraints imposed during adaptation balance the unavoidable tradeoff between the adapted template features not matching some relevant noise features (underfitting) versus matching some irrelevant signal features in the target dataset (overfitting).
Example applications of adapting template datasets include Surface multiple attenuation, ground roll removal, 4D seismic differencing, and Internal multiple attenuation. Next, some adaptation approaches that have been used and disclosed in the published literature are briefly described.
Least Squares Matching (and Subtraction for Noise Attenuation)
A commonly employed adaptation approach (see U.S. Pat. No. 6,894,948) called the least-squares method adapts the template within small data windows using constrained convolutional filters such that energy of the residual data after subtraction of the adapted template from the target dataset is minimized. In one embodiment, an upgoing wavefield from a seismic experiment is used as a template for the noise in the downgoing wavefield. The template is adapted by convolution with a set of filters so that it best matches the data in the least squares sense; that is, the energy in the difference between the signal and the adapted template is minimized.
This method is also applied widely in traditional noise suppression, where prediction of reverberations are first adaptively matched to reverberations in the data and then subtracted from seismic data. See, for example,    1. Target-oriented adaptive subtraction in data-driven multiple removal, Roald van Borselen, Grog Fookes, and John Brittan, The Leading Edge 22, 340 (2003)    2. Comparisons of adaptive subtraction techniques for multiple attenuation, Ray Abma, Nurul Kabir, Ken H. Matson, Simon A. Shaw, Bill McLain, and Scott Michell, SEG Expanded Abstracts 21, 2186 (2002)    3. Minimum energy adaptive subtraction in surface-related multiple attenuation, A. G. Nekut and D. J. Verschuur, SEG Expanded Abstracts 17, 1507 (1998)    4. Moveout-discriminating adaptive subtraction of multiples, Clement Kostov and Dave Nichols, SEG Expanded Abstracts 14, 1464 (1995)    5. Adaptive subtraction of emulated multiples, Philippe Julien and Jean-Jacques Raoult, SEG Expanded Abstracts 8, 1118 (1989)
The various current methods of adaptive matching attempt to use characteristics of the difference between signal and noise, plus the pattern or template of the noise obtained from a prediction method, to optimally match and subsequently remove the noise. Filters can be 1-D or 2-D (i.e. either only in time or in both time and space), and methods can use energy minimization in the window (after matching and subtraction) either weighted or not by prior information such as dip differences between signal and noise.
The least-squares based adaptation approach is limited by the rigid choice of adaptation constraints that is imposed on all reflectors that fall within a chosen data window. When such an adapted template is employed in the noise attenuation problem, the rigid choice of adaptation constraints within each window prevents the least squares approach from balancing the tradeoff between noise subtraction and signal preservation on a reflector-by-reflector basis.
Nonlinear Adaptive Matching and Removal of Template Patterns
Another approach is to use the template with the data in a nonlinear way in order to suppress the noise from the data. As disclosed in “Dereverberation of seismic data by 2-D nonlinear filtering: A wave equation-based approach,” Binzhong Zhou and Stewart Greenhalgh, SEG Expanded Abstracts 10, 1315 (1991), the template and the data are both either used in the time-space domain or transformed into a traditional domain such as 2-D Fourier, and the portions of the data that overlap the template in the chosen domain are removed or reduced in amplitude by a formula that may be nonlinear with respect to the data amplitudes. The approach matches the template to the data (i.e. allows for adaptation due to an imperfect prediction) by controlled smoothing or blurring of the template in the domain where the nonlinear reduction is applied. Among the limitations to this approach, blurring parameters must be chosen experimentally for each case. Since the method is nonlinear, the robustness of the method in preserving signal while reducing noise is not well-understood.
Curvelet-Based Adaptive Matching and Removal of Template Patterns
A recent approach implicitly shapes a template by employing a new data representation called a curvelet representation. This approach first expresses the target and template datasets using a weighted sum of real-valued curvelet functions, which resemble pieces of seismic reflectors. These weights are called curvelet coefficients; such weights are analogous to “Fourier coefficients” that serve as the weights of sinusoids in the representation of signals in the Fourier domain. The implicit adaptation modifies the magnitudes of the template's curvelet coefficients to better match the target's curvelet coefficients. The level of adaptation is controlled by constraining the magnitudes to vary only within a specific range.
A collection of related recent approaches (Hermann and Verschuur, 2004; Yarham et al, 2006; Beyreuther et al, 2005) disclose such a use of real-valued curvelet representations to implicitly adapt a template dataset and subtract it from the target. The adaptation is carried out by “shrinking” (that is, by reducing their magnitudes) the template's curvelet coefficients so that they match the target's curvelet coefficients better. The level of shrinkage is constrained by solely using the magnitudes of the template's curvelet coefficients. The applications addressed in (Hermann and Verschuur, 2004; Yarham et al, 2006; Beyreuther et al, 2005) include dereverberation or multiple noise attenuation, surface wave (ground roll) mitigation, and computation of time-lapse differences.